Showing papers by "Ming-Yang Kao published in 1999"

Linear-Time Succinct Encodings of Planar Graphs via Canonical Orderings

Abstract: Let G be an embedded planar undirected graph that has n vertices, m edges, and f faces but has no self-loop or multiple edge. If G is triangulated, we can encode it using 4/3m-1 bits, improving on the best previous bound of about 1.53m bits. In case exponential time is acceptable, roughly 1.08m bits have been known to suffice. If G is triconnected, we use at most $(2.5+2\log{3})\min\{n,f\}-7$ bits, which is at most 2.835m bits and smaller than the best previous bound of 3m bits. Both of our schemes take O(n) time for encoding and decoding.

A Decomposition Theorem for Maximum Weight Bipartite Matchings with Applications to Evolutionary Trees

Abstract: Let G be a bipartite graph with positive integer weights on the edges and without isolated nodes. Let n and W be the node count and the total weight of G. We present a new decomposition theorem for maximum weight bipartite matchings and use it to design an O(?nW)-time algorithm for computing a maximum weight matching of G. This algorithm bridges a long-standing gap between the best known time complexity of computing a maximum weight matching and that of computing a maximum cardinality matching. Given G and a maximum weight matching of G, we can further compute the weight of a maximum weight matching of G - {u} for all nodes u in O(W) time. As immediate applications of these algorithms, the best known time complexity of computing a maximum agreement subtree of two l-leaf rooted or unrooted evolutionary trees is reduced from O(l1.5 log l) to O(l1.5).

Optimal buy-and-hold strategies for financial markets with bounded daily returns

Abstract: In the context of investment analysis, we formulate an abstract online computing problem called a planning game and develop general tools for solving such a game. We then use the tools to investigate a practical buy-and-hold trading problem faced by long-term investors in stocks. We obtain the unique optimal static online algorithm for the problem and determine its exact competitive ratio. We also compare this algorithm with the popular dollar averaging strategy using actual market data.

Recovering evolutionary trees through harmonic greedy triplets

Abstract: We give a greedy learning algorithm for reconstructing an evolutionary tree based on a harmonic average on triplets of taxa. This algorithm runs in polynomial time in the input size. Using the Jukes-Cantor model of evolution, our algorithm is mathematically proven to require sample sequences of only polynomial lengths in the number of taxa in order to recover the correct tree topology with high probability. In addition to recovering the topology, the algorithm also estimates the tree edge lengths with high accuracy. Our theoretical analysis is supported by simulated experiments, in which the algorithm has demonstrated high success rates, in reconstructing a large tree from short sequences.

On-Line Difference Maximization

Abstract: In this paper we examine problems motivated by on-line financial problems and stochastic games. In particular, we consider a sequence of entirely arbitrary distinct values arriving in random order, and must devise strategies for selecting low values followed by high values in such a way as to maximize the expected gain in rank from low values to high values. First, we consider a scenario in which only one low value and one high value may be selected. We give an optimal on-line algorithm for this scenario, and analyze it to show that, surprisingly, the expected gain is n-O(1), and so differs from the best possible off-line gain by only a constant additive term (which is, in fact, fairly small---at most 15). In a second scenario, we allow multiple nonoverlapping low/high selections, where the total gain for our algorithm is the sum of the individual pair gains. We also give an optimal on-line algorithm for this problem, where the expected gain is $n^2/8-\Theta(n\log n)$. An analysis shows that the optimal expected off-line gain is $n^2/6+\Theta(1)$, so the performance of our on-line algorithm is within a factor of 3/4 of the best off-line strategy.

Balanced randomized tree splitting with applications to evolutionary tree constructions

Abstract: We present a new technique called balanced randomized tree splitting. It is useful in constructing unknown trees recursively. By applying it we obtain two new results on efficient construction of evolutionary trees: a new upper time-bound on the problem of constructing an evolutionary tree from experiments, and a relatively fast approximation algorithm for the maximum agreement subtree problem for binary trees for which the maximum number of leaves in an optimal solution is large. We also present new lower bounds for the problem of constructing an evolutionary tree from experiments and for the problem of constructing a tree from an ultrametric distance matrix.

Balanced randomized tree splitting with applications to evolutionary tree constructions

Abstract: We present a new technique called balanced randomized tree splitting. It is useful in constructing unknown trees recursively. By applying it we obtain two new results on efficient construction of evolutionary trees: a new upper time-bound on the problem of constructing an evolutionary tree from experiments, and a relatively fast approximation algorithm for the maximum agreement subtree problem for binary trees for which the maximum number of leaves in an optimal solution is large. We also present new lower bounds for the problem of constructing an evolutionary tree from experiments and for the problem of constructing a tree from an ultrametric distance matrix.

Optimal Bidding Algorithms Against Cheating in Multiple-Object Auctions

Abstract: This paper studies some basic problems in a multiple-object auction model using methodologies from theoretical computer science. We are especially concerned with situations where an adversary bidder knows the bidding algorithms of all the other bidders. In the two-bidder case, we derive an optimal randomized bidding algorithm, by which the disadvantaged bidder can procure at least half of the auction objects despite the adversary's a priori knowledge of his algorithm. In the general k-bidder case, if the number of objects is a multiple of k, an optimal randomized bidding algorithm is found. If the k -- 1 disadvantaged bidders employ that same algorithm, each of them can obtain at least 1/k of the objects regardless of the bidding algorithm the adversary uses. These two algorithms are based on closed-form solutions to certain multivariate probability distributions. In situations where a closed-form solution cannot be obtained, we study a restricted class of bidding algorithms as an approximation to desired optimal algorithms.

Reducing Randomness via Irrational Numbers

Abstract: We propose a general methodology for testing whether a given polynomial with integer coefficients is identically zero. The methodology evaluates the polynomial at efficiently computable approximations of suitable irrational points. In contrast to the classical technique of DeMillo, Lipton, Schwartz, and Zippel, this methodology can decrease the error probability by increasing the precision of the approximations instead of using more random bits. Consequently, randomized algorithms that use the classical technique can generally be improved using the new methodology. To demonstrate the methodology, we discuss two nontrivial applications. The first is to decide whether a graph has a perfect matching in parallel. Our new NC algorithm uses fewer random bits while doing less work than the previously best NC algorithm by Chari, Rohatgi, and Srinivasan. The second application is to test the equality of two multisets of integers. Our new algorithm improves upon the previously best algorithms by Blum and Kannan and can speed up their checking algorithm for sorting programs on a large range of inputs.

A Fast General Methodology for Information - Theoretically Optimal Encodings of Graphs

Abstract: We propose a fast methodology for encoding graphs with informationtheoretically minimum numbers of bits. The methodology is applicable to general classes of graphs; this paper focuses on simple planar graphs. Specifically, a graph with property ? is called a ?-graph. If ? satisfies certain properties, then an n-node ?-graph G can be encoded by a binary string X such that (1) G and X can be obtained from each other in O(n log n) time, and (2) X has at most s(n)+o(s(n)) bits for any function s(n) = ?(n) so that there are at most 2s(n)+o(s(n)) distinct n-node ?-graphs. Examples of such ? include all conjunctions of the following sets of properties: (1) G is a planar graph or a plane graph; (2) G is directed or undirected; (3) G is triangulated, triconnected, biconnected, merely connected, or not required to be connected; and (4) G has at most l1 (respectively, l2) distinct node (respectively, edge) labels. These examples are novel applications of small cycle separators of planar graphs and settle several problems that have been open since Tutte's census series were published in 1960's.

Linear-time succinct encodings of planar graphs via canonical orderings

Abstract: Let G be an embedded planar undirected graph that has n vertices, m edges, and f faces but has no self-loop or multiple edge. If G is triangulated, we can encode it using 4/3m-1 bits, improving on the best previous bound of about 1.53m bits. In case exponential time is acceptable, roughly 1.08m bits have been known to suffice. If G is triconnected, we use at most $(2.5+2\log{3})\min\{n,f\}-7$ bits, which is at most 2.835m bits and smaller than the best previous bound of 3m bits. Both of our schemes take O(n) time for encoding and decoding.

Nonplanar topological inference and political-map graphs

Abstract: Given a political map M, we define a mixed graph G where the vertices correspond to the districts on M, the arcs encode the inclusion relation between the districts, and the edges indicate the sharing of boundary points of the districts. We prove that such graphs can be recognized in nondeterministic polynomial time. We then prove that a natural subclass of such graphs can be recognized in O(na(n) log n) time, where n is the number of vertices in the input graph and a(n) is an inverse of Ackermann’s function. The main result is an O(ncr(n)logn)-time algorithm for the following problem II: Given a planar graph G = (V,E) and a partition Vl,Vz,..., V, of V, decide whether G has a planar embedding E such that for every Vi, there is a face Fi in & whose boundary intersects every connected component of the subgraph induced by Vi:,. This result also has an application in printed circuit board design.

On the Informational Asymmetry between Upper and Lower Bounds for Ultrametric Evolutionary Trees

Abstract: This paper addresses the informational asymmetry for constructing an ultrametric evolutionary tree from upper and lower bounds on pairwise distances between n given species. We show that the tallest ultrametric tree exists and can be constructed in O(n2) time, while the existence of the shortest ultrametric tree depends on whether the lower bounds are ultrametric. The tallest tree construction algorithm gives a very simple solution to the construction of an ultrametric tree. We also provide an efficient O(n2)-time algorithm for checking the uniqueness of an ultrametric tree, and study a query problem for testing whether an ultrametric tree satisfies both upper and lower bounds.

Designing proxies for stock market indices is computationally hard

Abstract: In this paper, we study the problem of designing proxies (or portfolios) for various stock market indices based on historical data. We use four different methods for computing market indices, all of which are formulae used in actual stock market analysis. For each index, we consider three criteria for designing the proxy: the proxy must either track the market index, outperform the market index, or perform within a margin of error of the index while maintaining a low volatility. In eleven of the twelve cases (all combinations of four indices with three criteria except the problem of sacrificing return for less volatility using the price-relative index) we show that the problem is NP-hard, and hence most likely intractable.

Linear-Time Approximation Algorithms for Computing Numerical Summation with Provably Small Errors

Abstract: Given a multiset $X=\{x_1,..., x_n\}$ of real numbers, the {\it floating-point set summation} problem asks for $S_n=x_1+...+x_n$. Let $E^*_n$ denote the minimum worst-case error over all possible orderings of evaluating $S_n$. We prove that if $X$ has both positive and negative numbers, it is NP-hard to compute $S_n$ with the worst-case error equal to $E^*_n$. We then give the first known polynomial-time approximation algorithm that has a provably small error for arbitrary $X$. Our algorithm incurs a worst-case error at most $2(\mix)E^*_n$.\footnote{All logarithms $\log$ in this paper are base 2.} After $X$ is sorted, it runs in O(n) time. For the case where $X$ is either all positive or all negative, we give another approximation algorithm with a worst-case error at most $\lceil\log\log n\rceil E^*_n$. Even for unsorted $X$, this algorithm runs in O(n) time. Previously, the best linear-time approximation algorithm had a worst-case error at most $\lceil\log n\rceil E^*_n$, while $E^*_n$ was known to be attainable in $O(n \log n)$ time using Huffman coding.